Making a Mathematical Statistics Course More Modern

USCOTS 2025

Jessica Chapman

St. Lawrence University (USA)

Matt Higham

St. Lawrence University (USA)

2025-07-18

About Us

  • Graduate School: Iowa State University; Undergrad: Truman State University.
  • Research: Multi-criteria optimization and STEM Education (NSF S-STEM).
  • Favorite Course to Teach: Probability and (increasingly) Math Stat.

About Us

  • Graduate School: Oregon State University; Undergraduate School: Miami University (OH).
  • Research: Spatial statistics and R package development.
  • Favorite Course to Teach: Foundations of Data Science.

Your Turn

Form groups of 3 to 5 people with those around you. Introduce yourself by discussing one or more of the following questions:

  1. What is your favorite course to teach? Do you have a favorite topic in that course? If so, what is it?
  2. If you could ban one statistical term forever, what would it be?
  3. What statistical concept do you secretly love?
04:00

Motivation for Session

  1. What more can we do to make math stat more “useful?”
  2. What topics should we be including in a math stat course? What topics do today’s students find interesting and engaging?
  3. How can we teach math stat in a way that students find useful, interesting, and engaging?

What Do We Mean by “Math-Stat”?

Motivation: Student Feedback

Feedback A

“I really only think this class is applicable to someone going to grad school or heavily love mathematics. The class is great in that it is difficult and extends your math but very low applicability to life/career.”

Feedback B

“There are so many components of this course that I will be able to apply to my future career and life experiences, a lot of which I think everyone regardless of field of study should at least know a little about.”

Your Turn

What does it mean for a math-stat course to be “useful?” In your groups, try to come up with a specific definition for “useful” in this context.

05:00

What Our Students Said (Top 3 Response Types)

  1. Applicable to Future: “I defined useful as something I think I will continue to use or see again in the future—skills or ideas that expand into future learning goals.”
  2. Applicable to the “Real World”: “I based this on how applicable these things would be to my professional life or general knowledge after college….”
  3. Deepened Understanding: “I (also) considered how much something deepened my conceptual understanding.”

Your Turn

What topics should be included in a math stat course? Consider a list of topics in an accompanying handout, and, with your groups, decide on 15 that you feel should be in a Math Stat course.

10:00

Your Turn

Have one member of your group fill out the following google form with your 15 math-stat “essentials.”

https://forms.gle/Y43VAEotVp1isKYn8

Post In or Out

  • JSM 2003 Session in San Francisco titled Is the Math Stat Course obsolete?
  • Panelists: Nancy Reid (Toronto), Bradley Efron (Stanford), Carl Morris (Harvard).
  • Chair: David Moore (Purdue).
  • “The math stat course has not changed in 40 years, whereas statistics has changed enormously, so how could the course not be obsolete?” - Moore

Post In or Out: Agree or Disagree?

Teach proofs? “I don’t know. My first reaction is to say do not teach formal derivation. \(\text{Var}(\bar{X})\) is reasonably accessible, but if Efron did not understand the t distribution (derivation) when he first saw it, then your students won’t. I can’t see emphasizing so much the derivations of these distributions, especially those using multivariate calculus. Reality has surpassed us - you will not be able to squeeze all of this into this course.” - Reid.

Post In or Out: Agree or Disagree?

“Leave out point estimation (or allow 5 min tops), hypothesis testing, power, type I error, type II error, and sample size calculation. Put these topics into the ”learn to be a consultant” course. Leave out one sample z, two sample t, one sample t, chi-square. Leave it out! There is a lot in the traditional course that is worth while but the psyche of the traditional course is not working anymore. The ways to turn it around are not that different than the statistical literacy course…..leave out UMVUE and unbiasedness.” - Reid

Post In or Out: Agree or Disagree?

Moore: “I heard no support for continuing to teach optimal testing (Neyman-Pearson) in a student’s first math-stat course?”

All: “Right.”

Post In or Out: Survey Results

Post In or Out: Bias and Variance

  1. Compute bias, variance, and MSE of an estimator “by hand” through integration.

  2. Approximate bias, variance, and MSE of an estimator computationally through Monte Carlo simulation.

  3. Construct plots of estimator PDFs that demonstrate conceptual understanding of “high bias, low variance” estimators and “low bias, high variance” estimators.

  4. Explain what it means for an estimator to have high variance.

  5. Extend concepts beyond a simple estimator \(\hat{\theta}\) to more complex settings (like a comparison of simple linear regression and Lasso regression).

Ideas for Course Updates

  1. Demonstrating depth of understanding of the concepts in an “estimation” section of math stat.
  2. Introducing Bayesian with an in-class activity.
  3. Discussing the benefits and drawbacks of using p-values to answer research questions.

Activity: Short Story

  • Fostering Conceptual Understanding in Mathematical Statistics by Green and Blankenship (2015).
    • published in The American Statistician.

Activity: Short Story

A “meaningful story” is one continuous piece of writing / creative work that uses key words from a list and in which the sentences “make sense and hang together.” That is, the ideas in the story must illustrate that you understand key concepts from math-stat in a way that allows you to write “meaningfully” about them. It is your job to use the terms in a way that demonstrates that you understand the statistical concepts involved and why we care about these terms in the big picture of statistical theory.

Your Turn

With your small group, orally construct a meaningful paragraph about a topic that you will draw from a hat using 3 of the following 5 terms:

  • Estimator
  • Estimate (as a noun)
  • Parameter
  • Bias
  • Variance
08:00

What Our Students Wrote About

  • 80’s music.
  • applying to jobs (this is a spring semester course of mostly seniors!).
  • gender inequality in women’s and men’s hockey.
  • living on a farm.
  • all kinds of sports!

Student Feedback

“The second mini-project was the most impactful assessment for me….This project was very different from much of the Math/Stat curriculum and a unique opportunity to show understanding of different terms. Being forced to use these terms in the context of a sentence helps to develop a better understanding of what they mean. It is difficult to explain terms like estimator, variance, or binomial distribution in a statistical context of a story without a good understanding of their meanings….”

Activity: Introducing Bayes

  • One topic that we both brought in to Math Stat is Bayesian Statistics.

Activity: Introducing Bayes

  • Let X = number of people who prefer cats over dogs. Assume X ~ Binomial(n,p), where p = the proportion of people that prefer cats.

Activity: Introducing Bayes

Before we collect any data, what values do you think are reasonable for p? How strongly do you believe that? Use the Beta distribution app to identify a probability model that matches these beliefs.

Activity: Introducing Bayes

How can we combine your prior knowledge about the preference of cats to dogs with the data that we collected?

Activity: Introducing Bayes

  • How good are you at the game of flip cup?
  • What do you think are reasonable values for the probability that you successfully flip the cup upside-down?

Activity: Introducing Bayes

  • Perform n = 10 trials of flip-cup and keep track of the number of times that you successfully flip the cup upside-down.

How can we combine your prior knowledge about your flip cup abilities with the data that you collected?

03:00

Student Feedback

“I really hope that I remember Bayes. I feel like this would be helpful for me to remember because I want to study epidemiology and public health. As we learn more about disease and track new diseases it will probably be useful to update our ideas with what we have learned.”

Student Feedback

“I struggled to understand the Bayesian unit at first. I didn’t really understand why we were doing this odd method where we made up stuff to start. I finally understood how the prior distribution affected the posterior, with the parameters acting like observations. We can think that our prior beliefs are worth some actual data, which will contribute to the data we have to create a potentially more accurate model.”

Student Feedback

p-value Paper Introduction

Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “p< 0.05”. The American Statistician, 73 (sup1), 1-19.

p-value Paper Introduction

“Uncertainty exists everywhere in research…..Significance tests and dichotomized p-values have turned many researchers into scientific snowbirds, trying to avoid dealing with uncertainty by escaping to a”happy place” where results are either statistically significant or not. In the real world, data provide a noisy signal. Variation, one of the causes of uncertainty, is everywhere. Exact replication is difficult to achieve. So it is time to get the right (statistical) gear and “move toward a greater acceptance of uncertainty and embracing of variation” (Gelman 2016).

p-value Paper Introduction

“Why is eliminating the use of p-values as a truth arbiter so hard? ‘The basic explanation is neither philosophical nor scientific, but sociologic; everyone uses them,’ says Goodman (2019). It’s the same reason we can use money. When everyone believes in something’s value, we can use it for real things; money for food, and p-values for knowledge claims, publication, funding, and promotion. It doesn’t matter if the p-value doesn’t mean what people think it means; it becomes valuable because of what it buys.”

Student Assignment

  1. Towards the end of Section 1, the authors say “As ‘statistical significance’ is used less, statistical thinking will be used more.” Elaborate on what you think the authors mean. Give some examples of what you think embodies “statistical thinking.”

  2. Section 2, third paragraph: The authors state “A label of statistical significance adds nothing to what is already conveyed by [p-value]; in fact, this dichotomization of p-values makes matters worse.” Elaborate on what you think the authors means.

“These p-values We’ve Been Using for 4 Years Are Flawed??”

Student Feedback

“I always considered p-values to be the best way to measure significance and did not question what a p-value really meant. The project taught me what statistical thinking is and to take into account more than just whether a p-value passes some threshold. In intro statistics, we are taught to follow specific steps and formulas without much background knowledge or reasoning. I am someone who does not often question information, especially in classes…..”

Your Turn

In your small groups, discuss one or more of the following:

  1. What have you done to make your math stat more modern?

  2. What ideas do you have about making your math stat more modern, either based from this session or from other sources?

07:00

Your Turn

If you were going to make one change to your math stat course, what would it be?

03:00

Projects and Labs

Thank You!

Modernizing The Undergraduate Stat. Theory Course